Null Energy Condition in Gravity

Updated 13 November 2025

Null Energy Condition (NEC) is a fundamental condition that requires Tμνk^μk^ν ≥ 0 for all null vectors, ensuring the physicality of energy in spacetime.
It underpins key gravitational results such as singularity theorems and the focusing of null geodesics, with implications in cosmology and effective field theories.
Quantum extensions like the ANEC, QNEC, and DSNEC offer controlled violations of the NEC, paving the way for non-singular cosmological models and novel exotic spacetime solutions.

The Null Energy Condition (NEC) is one of the fundamental energy conditions in classical and semiclassical gravity, constraining the stress–energy tensor and hence the causal and geometric structure of spacetime. It plays a central role in singularity theorems, theorems forbidding exotic spacetimes such as traversable wormholes and time machines, and the robustness of effective field theories at both the ultraviolet and infrared. Quantum and semiclassical considerations necessitate a generalization of the NEC, but its variants—such as the Averaged Null Energy Condition (ANEC), achronal ANEC, double-smeared NEC, and the Quantum Null Energy Condition (QNEC)—continue to structure the interplay of gravitational physics and field theory across scales.

1. Mathematical Formulation and Physical Meaning

The NEC states that for every null vector $k^\mu$ ,

$T_{\mu\nu}k^\mu k^\nu \geq 0.$

In a perfect fluid or homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology with metric $ds^2 = -dt^2 + a^2(t)d\vec{x}^2$ , the NEC translates into the inequality $\rho + p \geq 0$ , where $\rho$ is the total energy density and $p$ is the total pressure. Under Einstein's equations, this further leads to $\dot H \leq 0$ , where $H = \dot a / a$ is the Hubble parameter. The NEC is equivalent to the "null convergence condition" on the Ricci tensor: $R_{\mu\nu}k^\mu k^\nu \geq 0$ , which is the necessary condition for the focusing of null geodesics, as codified in the Raychaudhuri equation.

The NEC underpins classical results such as the Penrose singularity theorem: if the NEC holds everywhere in an expanding cosmological model or a collapsing star, the formation of generic curvature singularities is inevitable. By contrast, controlled violations of the NEC allow for the construction of non-singular bouncing or genesis cosmological scenarios in which curvature invariants remain finite and the scale factor never vanishes (Cai et al., 2022).

2. Classical, Averaged, and Quantum Energy Conditions

Classical Pointwise NEC: As formulated above, the NEC places a local restriction on the stress–energy tensor. However, quantum field theory (QFT) generically violates the pointwise NEC due to large local energy fluctuations, as in the Casimir effect or the Boulware vacuum (0705.3193, Kontou, 2015).

Averaged Null Energy Condition (ANEC): To accommodate quantum violations, one instead considers the ANEC: $\int_{-\infty}^{+\infty} T_{\mu\nu}k^\mu k^\nu\, d\lambda \geq 0,$ where the integral is taken along a complete null geodesic parametrized by affine parameter $\lambda$ . The ANEC is robust in ruling out exotic spacetime geometries (e.g., traversable wormholes, time machines, negative-mass objects) and is provable for minimally coupled, free quantum fields in small-curvature backgrounds, under suitable technical conditions (0705.3193, Kontou et al., 2012, Kontou, 2015, Kelly et al., 2014).

Achronal ANEC: The "achronal" version of ANEC (AANEC) restricts the integral to null geodesics that are achronal (no two points are timelike separated). Self-consistent semiclassical gravity, with causal backreaction, preserves the AANEC even in the presence of quantum field theoretic violations of the NEC on non-achronal geodesics (0705.3193, 0910.5751).

Double Smeared NEC (DSNEC) and Smeared NEC (SNEC): Stronger quantum violations at finite segments motivate further averaging, such as DSNEC, which smears $T_{--}$ over both null directions, thereby establishing a finite lower bound in free fields: $\int du\,dv\,f(u)g(v)\langle T_{--}(u,v,\vec{y}=0) \rangle \geq -B,$ where $f,g$ are chosen smooth profiles and $B$ is calculable (Fliss et al., 2021). The SNEC conjectures a semilocal lower bound for the null-projected stress–energy over a finite segment as $\int f_\tau^2(\lambda) T_{\mu\nu} k^\mu k^\nu d\lambda \geq -B/\tau^2$ , with explicit cosmological constraints on dark energy and bounce durations (Moghtaderi et al., 25 Mar 2025).

Quantum Null Energy Condition (QNEC): The QNEC replaces $T_{kk}\geq 0$ by a lower bound in terms of the second null variation of von Neumann entropy $S$ across the null congruence: $T_{kk}(p) \geq \frac{1}{2\pi} S''|_{\lambda=0}.$ This is a local, renormalization-scheme–independent condition in $d\leq3$ for vanishing expansion, and under stronger stationarity and energy conditions in $d=4,5$ (Fu et al., 2017, Ecker et al., 2017).

3. NEC Violation in Cosmology: Theoretical Realizations and Pathologies

NEC violation enables cosmological mechanisms not possible in standard GR:

Genesis/Bounce Cosmologies: Non-singular genesis or bounce solutions with $\dot H>0$ require controlled NEC violation. The explicit genesis solution in cubic Galileon theory is $\dot\phi(t) = M/(-t)$ with $a(t)\simeq 1+[(\lambda_2+\lambda_3)/(6M_P^2)]/(-t)^2$ for $t\ll-1/M$ , realizing $\dot H>0$ while approaching Minkowski at early times (Cai et al., 2022).
DBI Genesis: The DBI conformal galileon theory provides the first example of a Lagrangian with a stable Poincaré vacuum, a subluminal, radiatively stable NEC-violating cosmological solution, and a standard analytic S-matrix (Hinterbichler et al., 2012). Ghost and gradient instabilities are controlled via the unique combination of higher-derivative operators.
"Beyond Horndeski" Theories: The operator $\delta g^{00}R^{(3)}$ is essential for fully stable NEC violation at both early- and late-times, as it can render both the kinetic and gradient terms positive for curvature perturbations in the EFT of cosmology, overcoming the instabilities that plague standard scalar-tensor (Horndeski) models. Model-independent constraints show that the presence of this operator is favored by data indicating "phantom crossing" in dark energy (Ye et al., 28 Mar 2025).
Mimetic Cosmology: Mimetic gravity can violate the NEC at the level of the background, but it is generically plagued by gradient instabilities in the scalar perturbation sector, rooted in the structure of the reduced action—casting doubt on its viability except as a degenerate limit of higher-derivative healthy models (Ijjas et al., 2016).

4. Quantum Energy Inequalities, ANEC Proofs, and Holography

Quantum Inequalities: Flat-space quantum field theory provides lower bounds for energy densities smeared along timelike or null directions; these "quantum inequalities" generalize to curved space in small-curvature tubular neighborhoods, providing the foundational estimates for the (A)ANEC (Kontou, 2015, Kontou et al., 2012).

Proofs in Curved and Self-Consistent Settings: For a minimally coupled scalar in a classical background with small curvature, the (A)ANEC holds for all nearby geodesics. Achronicity is crucial—topological censorship and chronology protection are immediate consequences under achronal ANEC (0705.3193, 0910.5751, Kontou et al., 2012, Kontou, 2015).

Holographic Proofs: In AdS/CFT, the ANEC is equivalent to bulk causality—if violated, superluminal signaling in the boundary theory would result. The holographic proof proceeds via the Fefferman–Graham expansion for the near-boundary bulk metric. Boundary causality constraints—no causal curve starting and ending on the boundary can outpace the boundary light cone—can be translated to the ANEC for the expectation value in the CFT (Kelly et al., 2014).

5. NEC Violation: Observational, Cosmological, and Swampland Contexts

Observational Status: Current cosmological data robustly support the NEC for the total cosmic fluid—violations by individual fluid components (e.g., dark energy with $w<-1$ ) are neither necessary nor sufficient for NEC violation, and arise frequently from "phantom mirages" caused by sector misassignment or effective-fluid modeling (Caldwell et al., 10 Nov 2025). The NEC involves only the sum $\sum_i (\rho_i + P_i)$ , not individual species.

Smeared NEC and Cosmological Constraints: The smeared null energy condition imposes nontrivial bounds on parameter space for phantom dark energy ( $w<-1$ ), the duration and rapidity of cosmic bounces, and connecting cosmological epochs. For example, the allowed level of NEC breaking by dark energy today is tightly constrained by measurements of the Hubble parameter and the detailed expansion history (Moghtaderi et al., 25 Mar 2025).

Parity-Violating and Primordial Signatures: NEC violation correlates with unique phenomenological signatures—blue-tilted primordial GW spectra, peaks in the GW chirality parameter, and correlated PBH and SIGW signals—particularly when the NEC-violating background scalar directly couples to parity-violating gravity densities, as in Nieh–Yan–coupled cosmologies (Jiang et al., 24 Jun 2024, Cai et al., 2023).

UV Completion and the Swampland: The enforcement of the NEC is a necessary condition for the existence of a Wilsonian 4D effective field theory consistent with a UV-completion in M-theory, at least for closed or flat FLRW backgrounds, while NEC-violating models in these settings are in the swampland (Bernardo et al., 2021). Violation of the NEC causes the premature breakdown of the EFT hierarchy due to uncontrolled backward propagation of the string coupling in the power expansion, thereby spoiling the existence of a tractable low-energy theory. Notably, this does not impose the higher-dimensional NEC in M-theory; the constraint is four-dimensional and emerges from the imposed hierarchy structure.

6. Geometric, Theoretical, and Multicomponent Extensions

Bimetric Gravity and NEC Anticorrelation: In ghost-free bimetric gravity, there exist two interacting dynamical metrics. The NECs in the foreground and background sectors are strictly anti-correlated—one or the other must be violated except when their effective stress–energy tensors reduce to cosmological constants. This anti-correlation leads to novel opportunities for exotic matter-free wormhole solutions, superacceleration, or altered causal structures in one sector, subject to the overall stability and consistency of the theory (Baccetti et al., 2012).

Generalizations and Local/Nonlocal Conditions:

The QNEC stands as the only universally local quantum energy condition, with robust holographic and field-theoretic evidence in multiple dimensions (Fu et al., 2017, Ecker et al., 2017).
DSNEC and related worldvolume bounds provide a quantum-infused yet semilocal lower bound, enabling a generalization of singularity theorems and topological censorship in semi-classical gravity (Fliss et al., 2021).

Open Questions:

The scope of the NEC and its generalizations—in particular the final status of the self-consistent achronal ANEC and the precise determination of optimal constants in smeared quantum inequalities—remains a target for further research (0705.3193).
The conjunction of semiclassical, quantum, and swampland constraints continues to refine the permitted range of theories compatible with gravity and quantum field theory.

In summary, the Null Energy Condition and its extensions serve as structurally central, deeply interlinked constraints in gravitational theory, quantum field theory, and cosmology. Stable, controlled NEC violation is a nontrivial challenge at both the classical and quantum levels, yet remains essential for viable alternatives to standard cosmological scenarios, including nonsingular bounces, genesis models, and controlled instances of "phantom" dark energy. The interface of observational cosmology, EFT analysis, and UV-completeness criteria imposes increasingly stringent requirements, reinforcing the foundational role of the NEC and its quantum generalizations.